p?heevr

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 6/24/2024

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p?heevr

Computes selected eigenvalues and, optionally, eigenvectors of a Hermitian matrix using Relatively Robust Representation.

Syntax

call pcheevr( jobz, range, uplo, n, a, ia, ja, desca, vl, vu, il, iu, m, nz, w, z, iz, jz, descz, work, lwork, rwork, lrwork, iwork, liwork, info )

call pzheevr( jobz, range, uplo, n, a, ia, ja, desca, vl, vu, il, iu, m, nz, w, z, iz, jz, descz, work, lwork, rwork, lrwork, iwork, liwork, info )

Include Files

Description

p?heevr computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A distributed in 2D blockcyclic format by calling the recommended sequence of ScaLAPACK routines.

First, the matrix A is reduced to complex Hermitian tridiagonal form. Then, the eigenproblem is solved using the parallel MRRR algorithm. Last, if eigenvectors have been computed, a backtransformation is done.

Upon successful completion, each processor stores a copy of all computed eigenvalues in w. The eigenvector matrix Z is stored in 2D block-cyclic format distributed over all processors.

Product and Performance Information
Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex. Notice revision #20201201

Product and Performance Information

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.

Notice revision #20201201

Input Parameters

jobz

(global) CHARACTER*1

Specifies whether or not to compute the eigenvectors:

= 'N': Compute eigenvalues only.

= 'V': Compute eigenvalues and eigenvectors.

range

(global) CHARACTER*1

= 'A': all eigenvalues will be found.

= 'V': all eigenvalues in the interval [vl,vu] will be found.

= 'I': the il-th through iu-th eigenvalues will be found.

uplo

(global) CHARACTER*1

Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:

= 'U': Upper triangular

= 'L': Lower triangular

n

(global ) INTEGER

The number of rows and columns of the matrix A. n≥ 0

a

COMPLEX for pcheevr

COMPLEX*16 for pzheevr

Block-cyclic array, global size (n, n), local size ( lld_a, LOC_c(ja+n-1) )

Contains the local pieces of the Hermitian distributed matrix A. If uplo = 'U', only the upper triangular part of a is used to define the elements of the Hermitian matrix. If uplo = 'L', only the lower triangular part of a is used to define the elements of the Hermitian matrix.

ia

(global ) INTEGER

Global row index in the global matrix A that points to the beginning of the submatrix which is to be operated on. It should be set to 1 when operating on a full matrix.

ja

(global ) INTEGER

Global column index in the global matrix A that points to the beginning of the submatrix which is to be operated on. It should be set to 1 when operating on a full matrix.

desca

(global and local) INTEGER array of size dlen_. (The ScaLAPACK descriptor length is dlen_ = 9.)

The array descriptor for the distributed matrix a. The descriptor stores details about the 2D block-cyclic storage, see the notes below. If desca is incorrect, p?heevr cannot work correctly.

Also note the array alignment requirements specified below

vl

REAL for pcheevr

DOUBLE PRECISION for pzheevr

(global)

If range='V', the lower bound of the interval to be searched for eigenvalues. Not referenced if range = 'A' or 'I'.

vu

REAL for pcheevr

DOUBLE PRECISION for pzheevr

(global)

If range='V', the upper bound of the interval to be searched for eigenvalues. Not referenced if range = 'A' or 'I'.

il

(global ) INTEGER

If range='I', the index (from smallest to largest) of the smallest eigenvalue to be returned. il≥ 1.

Not referenced if range = 'A'.

iu

(global ) INTEGER

If range='I', the index (from smallest to largest) of the largest eigenvalue to be returned. min(il,n) ≤iu≤n.

Not referenced if range = 'A'.

iz

(global ) INTEGER

Global row index in the global matrix Z that points to the beginning of the submatrix which is to be operated on. It should be set to 1 when operating on a full matrix.

jz

(global ) INTEGER

Global column index in the global matrix Z that points to the beginning of the submatrix which is to be operated on. It should be set to 1 when operating on a full matrix.

descz

(global and local) INTEGER array of size dlen_.

The array descriptor for the distributed matrix z. descz( ctxt_ ) must equal desca( ctxt_ )

work

COMPLEX for pcheevr

COMPLEX*16 for pzheevr

(local workspace) array of size lwork

lwork

(local ) INTEGER

Size of work array, must be at least 3.

If only eigenvalues are requested:

lwork≥n + max( nb * ( np00 + 1 ), nb * 3 )

If eigenvectors are requested:

lwork≥n + ( np00 + mq00 + nb ) * nb

For definitions of np00 and mq00, see lrwork.

For optimal performance, greater workspace is needed, i.e.

lwork≥ max( lwork, nhetrd_lwork )

Where lwork is as defined above, and

nhetrd_lwork = n + 2*( anb+1 )*( 4*nps+2 ) + ( nps + 1 ) * nps

ictxt = desca( ctxt_ )

anb = pjlaenv( ictxt, 3, 'PCHETTRD', 'L', 0, 0, 0, 0 )

sqnpc = sqrt( real( nprow * npcol ) )

nps = max( numroc( n, 1, 0, 0, sqnpc ), 2*anb )

If lwork = -1, then lwork is global input and a workspace query is assumed; the routine only calculates the optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by pxerbla.

rwork

REAL for pcheevr

DOUBLE PRECISION for pzheevr

(local workspace) array of size lrwork

lrwork

(local ) INTEGER

Size of rwork, must be at least 3.

See below for definitions of variables used to define lrwork.

If no eigenvectors are requested (jobz = 'N') then

lrwork≥ 2 + 5 * n + max( 12 * n, nb * ( np00 + 1 ) )

If eigenvectors are requested (jobz = 'V' ) then the amount of workspace required is:

lrwork≥ 2 + 5 * n + max( 18*n, np00 * mq00 + 2 * nb * nb ) +

(2 + iceil( neig, nprow*npcol))*n

NOTE:

iceil(x,y) is the ceiling of x/y.

Variable definitions:

neig = number of eigenvectors requested

nb = desca( mb_ ) = desca( nb_ ) = descz( mb_ ) = descz( nb_ )

nn = max( n, nb, 2 )

desca( rsrc_ ) = desca( csrc_ ) = descz( rsrc_ ) = descz( csrc_ ) = 0

np00 = numroc( nn, nb, 0, 0, nprow )

mq00 = numroc( max( neig, nb, 2 ), nb, 0, 0, npcol )

iceil( x, y ) is a ScaLAPACK function returning ceiling(x/y), and nprow and npcol can be determined by calling the subroutine blacs_gridinfo.

If lrwork = -1, then lrwork is global input and a workspace query is assumed; the routine only calculates the size required for optimal performance for all work arrays. Each of these values is returned in the first entry of the corresponding work arrays, and no error message is issued by pxerbla

iwork

(local workspace) INTEGER array of size liwork

liwork

(local ) INTEGER

size of iwork

Let nnp = max( n, nprow*npcol + 1, 4 ). Then:

liwork≥ 12*nnp + 2*n when the eigenvectors are desired

liwork≥ 10*nnp + 2*n when only the eigenvalues have to be computed

If liwork = -1, then liwork is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by pxerbla

OUTPUT Parameters

a

The lower triangle (if uplo='L') or the upper triangle (if uplo='U') of a, including the diagonal, is destroyed.

m

(global ) INTEGER

Total number of eigenvalues found. 0 ≤m≤n.

nz

(global ) INTEGER

Total number of eigenvectors computed. 0 ≤nz≤m.

The number of columns of z that are filled.

If jobz≠ 'V', nz is not referenced.

If jobz = 'V', nz = m

w

REAL for pcheevr

DOUBLE PRECISION for pzheevr

(global ) array of size n

On normal exit, the first m entries contain the selected eigenvalues in ascending order.

z

COMPLEX for pcheevr

COMPLEX*16 for pzheevr

(local ) array, global size (n, n), local size ( lld_z, LOC_c(jz+n-1) )

If jobz = 'V', then on normal exit the first m columns of z contain the orthonormal eigenvectors of the matrix corresponding to the selected eigenvalues.

If jobz = 'N', then z is not referenced.

work

work(1) returns workspace adequate workspace to allow optimal performance.

rwork

On return, rwork(1) contains the optimal amount of workspace required for efficient execution. if jobz='N' rwork(1) = optimal amount of workspace required to compute the eigenvalues. if jobz='V' rwork(1) = optimal amount of workspace required to compute eigenvalues and eigenvectors.

iwork

On return, iwork(1) contains the amount of integer workspace required.

info

(global ) INTEGER

= 0: successful exit

< 0: If the i-th argument is an array and the j-th entry had an illegal value, then info = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then info = -i.

Application Notes

The distributed submatrices a(ia:*, ja:*) and z(iz:iz+m-1,jz:jz+n-1) must satisfy the following alignment properties:

Identical (quadratic) dimension: desca(m_) = descz(m_) = desca(n_) = descz(n_)
Quadratic conformal blocking: desca(mb_) = desca(nb_) = descz(mb_) = descz(nb_), desca(rsrc_) = descz(rsrc_)
mod( ia-1, mb_a ) = mod( iz-1, mb_z ) = 0

NOTE:

mod(x,y) is the integer remainder of x/y.

Parent topic: ScaLAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

p?heevr

Syntax

Include Files

Description

Input Parameters

OUTPUT Parameters

Application Notes

See Also